Solve $x^2-2 x=15$ by applying the quadratic formula.
A. $x=1+\sqrt{14} i, x=1-\sqrt{14} i$
B. $x=5, x=-3$
C. $x=-1+\sqrt{14} i, x=-1-\sqrt{14} i$
D. $x=-5, x=3$
Answer (2)
Rewrite the equation in standard form: x 2 − 2 x − 15 = 0 .
Identify the coefficients: a = 1 , b = − 2 , c = − 15 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Calculate the solutions: x = 5 and x = − 3 . The final answer is x = 5 , x = − 3 .
Explanation
Rewrite the equation We are given the quadratic equation x 2 − 2 x = 15 and asked to solve it using the quadratic formula. First, we need to rewrite the equation in the standard form a x 2 + b x + c = 0 .
Identify coefficients Subtracting 15 from both sides, we get x 2 − 2 x − 15 = 0 . Now we can identify the coefficients: a = 1 , b = − 2 , and c = − 15 .
Apply the quadratic formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c . Plugging in the values of a , b , and c , we have
x = 2 ( 1 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 1 ) ( − 15 )
x = 2 2 ± 4 + 60
x = 2 2 ± 64
x = 2 2 ± 8
Calculate the solutions Now we can find the two solutions:
x 1 = 2 2 + 8 = 2 10 = 5
x 2 = 2 2 − 8 = 2 − 6 = − 3
State the solutions Therefore, the solutions are x = 5 and x = − 3 .
Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For example, if you throw a ball, its path can be modeled by a quadratic equation, and solving the equation can help you determine how far the ball will travel.
To solve the equation x 2 − 2 x = 15 , we rewrite it as x 2 − 2 x − 15 = 0 and apply the quadratic formula. This results in two solutions: x = 5 and x = − 3 , so the correct answer is B. x = 5 , x = − 3 .
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